QUESTIONS AND PROBLEMS 1556.7. Verify that the solution to Problem 71 (2 3) is the correct pesu of the diluted
beer discussed in the problem.
6.8. Compare the pesu problems in the Ahmose Papyrus with the following prob-
lem, which might have been taken from almost any algebra book written in the past
century: A radiator is filled with 16 quarts of a 10% alcohol solution. If it requires
a 30% alcohol solution to protect the radiator from freezing when it is turned off,
how much 95% solution must be added (after an equal amount of the 10% solution
is drained off) to provide this protection? Think of the alcohol as the grain in beer
and the liquid in the radiator as the beer. The liquid has a pesu of 10. What is the
pesu that it needs to have, and what is the pesu of the liquid that is to be used to
achieve this result?
6.9. Verify that the solution 5 10 given above for Problem 35 is correct, that is,
multiply this number by 3 and by 3 and verify that the sum of the two results is 1.
6.10. Why do you suppose that the author of the Ahmose Papyrus did not choose
to say that the double of the thirteenth part is the seventh part plus the ninety-first
part, that is,
1 = 1 + 1?
13 7 T 91 ·
Why is the relation
13 8 T 52 T 104
made the basis for the tabular entry instead?
6.11. Generalizing Question 6.10, investigate the possibility of using the identity
2 _ 1 1
~Ñ~(Ø)- Ñ(Ø)
to express the double of the reciprocal of an odd number ñ as a sum of two recipro-
cals. Which of the entries in the table of Fig. 1 can be obtained from this pattern?
Why was it not used to express —º
6.12. Why not simply write 13 13 to stand for what we call ^? What is the reason
for using two or three other "parts" instead of these two obvious parts?
6.13. Could the ability to solve a problem such as Problem 35, discussed in Subsec-
tion 1.2 of this chapter, have been of any practical use? Try to think of a situation
in which such a problem might arise.
6.14. We would naturally solve many of the problems in the Ahmose Papyrus using
an equation. Would it be appropriate to say that the Egyptians solved equations,
or that they did algebra? What does the word algebra mean to you? How can you
decide whether you are performing algebra or arithmetic?
6.15. Why did the Egyptians usually begin the process of division by multiplying
by 3 instead of the seemingly simpler 2?6.16. Early mathematicians must have been adept at thinking in terms of expres-
sions. But considering the solutions to the riders-and-carts problem and the colorful
language of Brahmagupta in relation to the Rule of Three, one might look at the
situation from a different point of view. Perhaps these early mathematicians were
good "dramatists." In any algorithm the objects we now call variables amount to
special "roles" played, with different numbers being assigned to "act" in those roles;